Theorem ssrabdv 3236

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)

Hypotheses

Ref	Expression
ssrabdv.1	|- ( ph -> B C_ A )
ssrabdv.2	|- ( ( ph /\ x e. B ) -> ps )

Assertion

Ref	Expression
ssrabdv	|- ( ph -> B C_ { x e. A | ps } )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem ssrabdv

Step	Hyp	Ref	Expression
1		ssrabdv.1	. 2 |- ( ph -> B C_ A )
2		ssrabdv.2	. . 3 |- ( ( ph /\ x e. B ) -> ps )
3	2	ralrimiva 2550	. 2 |- ( ph -> A. x e. B ps )
4		ssrab 3235	. 2 |- ( B C_ { x e. A | ps } <-> ( B C_ A /\ A. x e. B ps ) )
5	1, 3, 4	sylanbrc 417	1 |- ( ph -> B C_ { x e. A | ps } )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   A.wral 2455   {crab 2459   C_ wss 3131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-in 3137  df-ss 3144

This theorem is referenced by: (None)